FairRARI: A Plug and Play Framework for Fairness-Aware PageRank

PageRank (PR) is a fundamental algorithm in graph machine learning tasks. Owing to the increasing importance of algorithmic fairness, we consider the problem of computing PR vectors subject to various group-fairness criteria based on sensitive attributes of the vertices. At present, principled algorithms for this problem are lacking - some cannot guarantee that a target fairness level is achieved, while others do not feature optimality guarantees. In order to overcome these shortcomings, we put forth a unified in-processing convex optimization framework, termed FairRARI, for tackling different group-fairness criteria in a ``plug and play’’ fashion. Leveraging a variational formulation of PR, the framework computes fair PR vectors by solving a strongly convex optimization problem with fairness constraints, thereby ensuring that a target fairness level is achieved. We further introduce three different fairness criteria which can be efficiently tackled using FairRARI to compute fair PR vectors with the same asymptotic time-complexity as the original PR algorithm. Extensive experiments on real-world datasets showcase that FairRARI outperforms existing methods in terms of utility, while achieving the desired fairness levels across multiple vertex groups; thereby highlighting its effectiveness.

💡 Research Summary

The paper tackles the pressing issue of fairness in PageRank (PR), a cornerstone algorithm for graph‑based centrality and ranking. Existing fairness‑aware PR methods either modify the teleportation vector, re‑weight transition probabilities, or alter the graph structure, but they lack provable guarantees of achieving a prescribed fairness level, do not provide optimality guarantees, and often cannot handle more than two demographic groups.

FairRARI (Fairness‑Aware Rank Iteration) introduces a unified in‑processing framework that embeds fairness constraints directly into the PR computation via a variational formulation. The authors first show that the standard PR vector is the unique minimizer of a strongly convex quadratic function
(f(x)=\frac{1}{2}(1-\gamma)x^{\top}\Pi^{-1}L_{rw}x+\frac{1}{2}\gamma|\Pi^{-1/2}(x-v)|^{2}),
where (L_{rw}=I-P) is the random‑walk Laplacian, (\Pi) encodes the stationary distribution, and (v) is the teleportation vector (uniform in the experiments). This formulation holds for both undirected and directed graphs (assuming time‑reversibility for the latter).

Fairness requirements are expressed as a closed convex set (X\subset\mathbb{R}^{n}). By solving the constrained problem (\min_{x\in X} f(x)) the algorithm guarantees that the resulting vector satisfies the chosen fairness criterion while remaining optimal with respect to the original PR objective. Because (f) is strongly convex, the solution is unique.

The algorithm proceeds in two simple steps per iteration:

1. PR update – a standard PageRank iteration: (y^{(t+1)}=(1-\gamma)Px^{(t)}+\gamma v).
2. Projection – Euclidean projection of (y^{(t+1)}) onto the fairness set: (x^{(t+1)}=\operatorname{Proj}_{X}(y^{(t+1)})).

The authors prove two key theorems: (i) for any convex set (X) the iteration converges geometrically to a unique fixed point, and (ii) this fixed point coincides exactly with the minimizer of the constrained problem. Thus the method can be interpreted as a preconditioned projected gradient descent where the PR step is a gradient step on (f) and the projection enforces feasibility.

Three fairness criteria are introduced:

• ϕ‑sum fairness – enforces that the total PR mass allocated to each protected group equals a prescribed fraction ϕ. This generalizes the notion used in prior work but now works for any ϕ∈(0,1).
• Minimum‑score fairness – guarantees a lower bound on the smallest PR score within each group, preventing degenerate solutions where many vertices receive zero score.
• Combined fairness – a linear combination of the above two, allowing fine‑grained control over both mass distribution and intra‑group equity.

For each criterion the authors devise a linear‑time projection algorithm, ensuring that the overall asymptotic complexity remains O(m), the same as vanilla PageRank. Importantly, the projection does not require pre‑computing the stationary distribution π, even though π appears in the objective; the iteration itself implicitly handles it.

Extensive experiments on real‑world graphs (e.g., POLBOOKS, citation networks, social networks) compare FairRARI against post‑processing baselines, LFPR variants, FairWalk, and other recent methods. Evaluation metrics include: (a) deviation from the target ϕ, (b) L2 distance between the fair and original PR vectors, (c) ranking quality (NDCG, Precision@k), and (d) the proportion of nodes receiving zero score. Results show that FairRARI consistently meets the exact fairness target, achieves lower utility loss than competitors, and dramatically reduces the number of zero‑score nodes—especially under the minimum‑score fairness condition. The framework also scales gracefully to multiple groups (more than two), a scenario where prior methods often fail.

In summary, FairRARI delivers a theoretically sound, plug‑and‑play solution for fairness‑aware PageRank. Its strengths lie in (1) a variational strong‑convex formulation, (2) a generic convex‑set projection mechanism that accommodates diverse fairness notions, (3) provable convergence to the unique optimal fair solution, and (4) unchanged computational complexity relative to standard PageRank. This makes it immediately applicable to any system that relies on PageRank‑style centrality—search engines, recommendation systems, influence maximization, and beyond—where equitable treatment of demographic groups is a mandatory requirement.